Explicit instruction with cumulative practice is a teaching approach commonly used in
mathematics to promote strong learning and retention. It involves two key elements:
Explicit Instruction:
• This refers to direct teaching where the teacher clearly and systematically
explains the concept being taught.
• The steps are broken down into manageable chunks, and the teacher provides
detailed explanations, examples, and non-examples to clarify the concept.
• Teachers often model the process first, showing students how to solve a problem
step-by-step.
• The key focus is on clear communication, structured guidance, and immediate
feedback to ensure students understand the material.
• Teachers also check for understanding throughout the lesson, ensuring students
are following along before moving on to more complex tasks.
Cumulative Practice:
• This refers to revisiting and practicing previous skills and concepts in addition to
learning new material.
• Cumulative practice helps reinforce learning and build connections between
different mathematical concepts.
• As students move forward, they repeatedly engage with earlier topics to
strengthen their understanding and retention.
• This ongoing review ensures that earlier concepts are not forgotten and that
students can apply them in new contexts or in combination with new learning.
• Cumulative practice might involve daily reviews or progressively more
challenging problems that combine past and current material.
How they work together in teaching mathematics:
• Explicit instruction helps students understand new content, while cumulative
practice ensures that this content is retained and applied effectively.
• For example, if you're teaching algebraic equations, you might first explicitly
teach how to solve one-step equations, then gradually add more complexity (like
two-step equations). After introducing new concepts, you would frequently review
solving one-step equations to keep skills fresh.
• Over time, students will be able to recall and apply previously learned concepts
automatically, which is crucial in mathematics where concepts build on each
other.
This combination of explicit teaching and cumulative practice is effective because it
supports both mastery of individual skills and the ability to apply those skills in more
complex, integrated (combined, merged with other field) problems. It also ensures that
students don’t struggle with foundational concepts (important ideas andprinciples) as
they advance, making the learning process more coherent (systematic/orderly) and
connected.
Examples:
Explicit Instruction
Objective: Students will learn how to add and subtract fractions with the same
denominator.
Step-by-Step Explanation:
Step 1: Explain the concept
"When you add or subtract fractions, the bottom number (the denominator) tells
us what parts the whole is divided into. If the fractions have the same
denominator, we only need to focus on the top numbers (the numerators)."
Step 2: Model an example of addition
1 2
"Let’s add + . The denominators are the same, so we keep the denominator
4 4
1 2 3
as 4. Now, add the numerators: 1 + 2 = 3. So, 4 + 4 = 4 .”
Step 3: Guided Practice
Now, check if the students are able to understand the concept. If they did, have
the students practice together. Write a few problems on the board and solve
them as a class.
As the teacher, you would walk through each step, ensuring that students understand
why the denominators remain the same and how to add or subtract the numerators.
Cumulative Practice
Objective: Students will practice previous skills to strengthen their understanding of
fractions, as well as apply the new skill to similar problems.
Step 1: Quick Review
Before jumping into the new content, briefly review earlier lessons where
students learned to identify and add fractions with the same denominator.
Step 2: Incorporate Earlier Concepts
Throughout the lesson, continuously connect the new material to prior
knowledge. For example, after the explicit instruction of adding fractions, remind
students how finding a common denominator works (even though they don't need
to do this for today's lesson, it’s a key concept for future lessons with different
denominators).
Step 3: Daily Cumulative Practice
To reinforce previous lessons, you can set aside a few minutes every day for
cumulative review:
o Solve a couple of simpler problems from earlier topics (e.g.,
adding/subtracting whole numbers and fractions)
o Gradually increase the complexity by combining previous skills, like asking
students to add or subtract fractions with like denominators.
Step 4: Integrating New Lesson
Give students few examples of the previous lesson and slowly introduce the new
discussion.
o For example, re-use the first equation you introduced in adding fractions
with common denominators.
1 2
+
4 4
o Change, alter the numbers as you like to get a possible answer that could
be simplified.
2 2
+4
4
o Let the students solve the equation.
2 2 4
+4=4
4
o Model the simplifying process. Discuss that in simplifying, finding the GCF
of both the numerator and denominator comes first.
4 4
÷4=1
4
o Check their understanding. Provide few more examples then, do
independent practice.
Putting it All Together:
• During explicit instruction, the teacher introduces the new concept and provides
clear, structured examples with explanations.
• In cumulative practice, the teacher integrates review of prior skills, offering
opportunities for students to recall and apply earlier knowledge while
solidifying new learning.
• The use of guided practice ensures that students receive ongoing support to
check for understanding, and the independent practice (which might follow
in a homework assignment or individual work) ensures that students are
applying both new and old knowledge.
This method promotes a coherent progression where new information is anchored in
prior learning, making it easier for students to build on and retain mathematical
concepts.
